Is it possible for a category to have an initial object $i$ such that

$$\text{Hom}(a,i) \neq \emptyset, \forall a \in \text{Obj}$$

And also such that this initial object is not a final object? I know that the empty set is an initial object but not a terminal object in $\text{Set}$, but that's mainly due to the fact that there are no arrows terminating at the empty set at all right?


1 Answer 1


Yes, this is possible. For instance, consider the opposite category of nonempty sets. A singleton set $i$ is initial in this category (since it is terminal in the usual category of sets), but there is a map from every object to $i$ (since there is a function from $i$ to any nonempty set). And $i$ is not terminal, since if $a$ is any set with more than one element then there is more than one map $a\to i$.

However, if you assume that your category has a terminal object $t$ and an initial object $i$ with a map from every object, then $i$ must be terminal. For if $f:t\to i$ is any map, let $g:i\to t$ be the unique map (using the universal property of either $i$ or $t$). Then $fg$ and $gf$ must both be the identity, because there is only one map $i\to i$ and only one map $t\to t$. So $f$ and $g$ are inverse isomorphisms, so $i$ is terminal since $t$ is.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .